Wednesdays at 4 PM in Eckhart 202.
We show that, in the hydrodynamic limit, the Heisenberg dynamics of the energy, momentum, and particle densities for fermions with short-range pair interactions converges to the compressible Euler equations with the pressure function given by quantum statistical mechanics. Our derivation is based on a quantum version of the entropy method and a suitable quantum virial theorem. We require a number of technical conditions that will be discussed in the talk (joint work with H.T. Yau).
We investigate two-dimensional pattern formation in a non-local neural model of Wilson-Cowan type. The first step in the analysis is to consider the scalar case and derive an equivalent partial differential equation. We then show how instability of axially symmetric solutions leads to the formation of multibump solutions. In the second part of the talk we extend the model to a system by including a recovery variable. In the system the formation of spiral waves has led Jian-Young Wu(Georgetown U.) to search for spirals in tangential slices of rat cortex. Videos will be shown which illustrate spirals in the model and also the experimentally observed spiral waves recently found by Wu's group. Finally, we will illustrate the formation of ring shaped waves which result from a stimulus applied to the center of the medium. Tangential slices of rat cortex have recently been shown to exhibit similar ring waves and these will also be illustrated.
The convective Cahn-Hilliard equation, u_t+(u_xx+u-u^3)_xx-Duu_x=0, has been suggested recently for the description of several physical phenomena, including spinodal decomposition of phase separating systems in an external field, step instability on a crystal surface, and faceting of thermodynamically unstable surfaces. This equation provides a "bridge" between the standard Cahn-Hilliard equation and the Kuramoto-Sivashinsky equation. Depending on the value of the parameter D, either domain coarsening (I) or the formation of regular or irregular patterns takes place. In the case (I), we investigate the dynamics of kinks that governs the domain coarsening. The important role of kink-antikink pairs and kink triplets is revealed. In the case (II), we investigate the stability of stationary and traveling-wave solutions.
We discuss a class of dispersive nonlinear nonlocal partial differential equations (PDEs) modeling mechanics and neural functions of the inner ear, then show analytical and numerical properties of wave like solutions under single and multiple frequency sound inputs. An intriguing nonlinear phenomenon is masking, namely, an audible sound becomes inaudible in the presence of another sound. We present a PDE based two level psychoacoustic model on masking of banded noise, and relate it to perceptual coding in digital music (MP3).
For questions, contact Eduard Kirr at email@example.com
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